A spectral characterization for concentration of the cover time

TL;DR

This paper establishes a spectral criterion for the concentration of cover times in finite graphs and reversible Markov chains, linking spectral gap and expected cover time to concentration behavior.

Contribution

It provides a new spectral characterization for cover time concentration applicable to general reversible Markov chains with mild assumptions.

Findings

Cover times are concentrated iff the spectral-gap times expected cover time diverges.

The result applies to vertex-transitive graphs and more general reversible chains.

The condition is weaker than transitivity, involving maximal and average hitting times.

Abstract

We prove that for a sequence of finite vertex-transitive graphs of increasing sizes, the cover times are asymptotically concentrated if and only if the product of the spectral-gap and the expected cover time diverges. In fact, we prove this for general reversible Markov chains under the much weaker assumption (than transitivity) that the maximal hitting time of a state is of the same order as the average hitting time.